Exponents

When powers with the same base are divided, their exponents are subtracted: \(a^m \div a^n=a^{m-n}\). Therefore, \(3^7 \div 3^3=3^{7-3}=3^4=81\). Hence, option C is correct. Exam tip: subtract exponents in division, whereas exponents are added when multiplying powers with the same base; therefore, 27 and 243 are not correct.

Given \(x=2^3=8\). Therefore, \(x^2=8^2=64\), so option B is correct. Note that \(x^2\) means squaring the complete value of \(x\), not squaring only the base 2 in \(2^3\). Exam tip: Using the exponent rule, \((2^3)^2=2^{3\times2}=2^6=64\) gives the answer directly.

When powers with the same non-zero base are divided, their exponents are subtracted: \(10^4 \div 10^2 = 10^{4-2}=10^2=100\). Therefore, the correct answer is 100. Exam tip: subtract exponents when dividing like bases; do not add them.

When powers with the same base are multiplied, their exponents are added: \(4^3 \times 4^2=4^{3+2}=4^5\). Since \(4^5=1024\), the correct answer is 1024. Exam tip: for \(a^m \times a^n\), use \(a^{m+n}\); do not multiply the exponents.

The zeroth power of any non-zero number is 1, so 7^0=1. Also, 7^1=7. Therefore, (7^0+7^1)=1+7=8, making option C correct. Exam tip: evaluate each power first and then perform the indicated addition; do not add the exponents directly.

Using the quotient rule for equal exponents, \(9^2 \div 3^2 = (9 \div 3)^2 = 3^2 = 9\). Therefore, the correct answer is 9. Exam tip: \(a^n \div b^n = (a \div b)^n\) can be applied when the exponents are equal.

For powers with the same base, exponents are added during multiplication and subtracted during division. Therefore, \((2^4 \times 2^3)\div 2^5 = 2^{4+3-5}=2^2=4\). Hence, option B is correct. Exam tip: Apply the laws of exponents before evaluating the final power.

By the zero-exponent rule, \(a^0=1\) for every non-zero number, so \(6^0=1\). Therefore, \(6^2 \times 6^0=36 \times 1=36\). Option B is incorrect because it does not evaluate \(6^2\). Exam tip: the zero power of any non-zero number is always 1.

Evaluate the powers first: \(8^2=64\) and \(2^5=32\). Therefore, \(8^2-2^5=64-32=32\), so option C is correct. Remember that exponentiation is performed before subtraction.

First evaluate the powers: \(3^4=81\) and \(3^2=9\). Therefore, \(3^4+3^2=81+9=90\), so option B is correct. Remember that powers cannot be added directly in an addition expression; evaluate each power separately and then add the results.

Since \(2^6=64\) and \(8=2^3\), we get \(8^2=(2^3)^2=2^6=64\). Therefore, option B is correct. Option A gives \(4^2=2^4\), while options C and D are much larger. In the exam, use the exponent rule \((a^m)^n=a^{mn}\).

When powers with the same base are divided, their exponents are subtracted: \(5^3 \div 5^1=5^{3-1}=5^2=25\). Therefore, the correct answer is 25. Remember to write the divisor 5 as \(5^1\) before applying the exponent rule.

Using the quotient rule for equal exponents, \\(a^n \div b^n = (a \div b)^n\\). Thus, \\(12^2 \div 6^2 = (12 \div 6)^2 = 2^2 = 4\\). Therefore, the correct answer is 4. Exam tip: do not divide only by 6; both the base and its square must be handled according to the exponent rule.

The first power of a non-zero number is the number itself, so \(11^1=11\). Also, the zero power of any non-zero number is 1, so \(11^0=1\). Therefore, \(11^1+11^0=11+1=12\). Exam tip: remember that \(a^0=1\) for \(a\neq0\).

Using the product rule for equal exponents, \(a^n \times b^n=(ab)^n\). Thus, \(2^3 \times 5^3=(2\times5)^3=10^3=1000\), so option B is correct. Exam tip: When the exponents are equal, multiply the bases first and then apply the common exponent.

First evaluate the powers: 7^2 = 49 and 2^5 = 32. Adding them gives 49 + 32 = 81, so option B is correct. Exam tip: evaluate exponents before performing the addition.

Evaluate the powers first: \(13^2=169\) and \(5^2=25\). Therefore, \(13^2-5^2=169-25=144\), so option A is correct. Exam tip: calculate both squares before subtracting; option B is only the value of \(13^2\), not of the complete expression.

Since \(a^n\times a^{-n}=a^{n-n}=a^0=1\), \(a^{-n}\) is the reciprocal of \(a^n\). Hence it equals \(\frac{1}{a^n}\), not \(-a^n\). Exam tip: convert a negative exponent into a reciprocal first.

Using the law \(\frac{a^n}{b^n}=\left(\frac{a}{b}\right)^n\) for equal exponents, \(\frac{15^2}{5^2}=\left(\frac{15}{5}\right)^2=3^2=9\). Therefore, 9 is correct. Exam tip: When the exponents are the same, divide the bases first and then apply the exponent.

For powers with the same base, the exponents are added: \(2^3\times2^2=2^{3+2}=2^5\). Then \((2^5)^2=2^{5\times2}=2^{10}=1024\). Therefore, option B is correct. Option C, 512, equals \(2^9\) and may result from applying the outer exponent incorrectly. Exam tip: remember \(a^m\times a^n=a^{m+n}\) and \((a^m)^n=a^{mn}\).

First evaluate the two powers: \(3^5=243\) and \(2^5=32\). Therefore, \(3^5+2^5=243+32=275\), so option D is correct. Option B is only the value of \(3^5\), not the complete expression. Exam tip: Evaluate each power separately before performing the addition.

Using the quotient rule for equal powers, \((25^2 \div 5^2)=(25\div 5)^2=5^2=25\). Therefore, option B is correct. Option A is only the quotient of the bases and misses the required squaring step. In an exam, remember the rule \(a^n\div b^n=(a\div b)^n\) when the exponents are equal.

Evaluate the powers first: 2^4=16 and 4^3=64. Therefore, 2^4+4^3=16+64=80, so option C is correct. In an exam, remember to calculate each power before performing the addition; 72 may result from an error in evaluating the powers.

First calculate the two squares: 14^2 = 196 and 10^2 = 100. Therefore, 14^2 - 10^2 = 196 - 100 = 96. This can also be verified using the difference-of-squares identity: a^2 - b^2 = (a-b)(a+b) = (14-10)(14+10) = 4 × 24 = 96. Exam tip: Use the difference-of-squares identity to simplify such expressions quickly.

Using the quotient rule for equal exponents, \(a^n \div b^n=(a\div b)^n\). Therefore, \(8^3\div2^3=(8\div2)^3=4^3=64\). Hence, 64 is correct. Exam tip: do not confuse \(8^3\div2^3\) with \(8\div2^3\); the placement of the exponent changes the expression.